8.1 Definition of the Integral

If $f$ is a monotonic function from an interval $[a,b]$ to $\mbox{{\bf R}}_{\geq 0}$, then we have shown that for every sequence $\{P_n\}$ of partitions on $[a,b]$ such that $\{\mu (P_n)\}\to 0$, and every sequence $\{S_n\}$ such that for all $n\in\mbox{${\mbox{{\bf Z}}}^{+}$}$ $S_n$ is a sample for $P_n$, we have

$$\{\sum (f,P_n,S_n)\}\to A_a^bf.$$

8.1   Definition (Integral.) Let $f$ be a bounded function from an interval $[a,b]$ to $\mbox{{\bf R}}$. We say that $f$ is integrable on $[a,b]$ if there is a number $V$ such that for every sequence of partitions $\{P_n\}$ on $[a,b]$ such that $\{\mu (P_n)\}\to 0$, and every sequence $\{S_n\}$ where $S_n$ is a sample for $P_n$

$$\{\sum (f,P_n,S_n)\}\to V.$$

If $f$ is integrable on $[a,b]$ then the number $V$ just described is denoted by $${\int_a^bf}$$ and is called `` the integral from $a$ to $b$ of $f$.'' Notice that by our definition an integrable function is necessarily bounded.

The definition just given is essentially due to Bernhard Riemann(1826-1866), and first appeared around 1860[39, pages 239 ff]. The symbol $${\int}$$ was introduced by Leibniz sometime around 1675[15, vol 2, p242]. The symbol is a form of the letter s, standing for sum (in Latin as well as in English.) The practice of attaching the limits $a$ and $b$ to the integral sign was introduced by Joseph Fourier around 1820. Before this time the limits were usually indicated by words.

We can now restate theorems 7.6 and 7.15 as follows:

8.2   Theorem (Monotonic functions are integrable I.) If $f$ is a monotonic function on an interval $[a,b]$ with non-negative values, then $f$ is integrable on $[a,b]$ and

$$\int_a^bf=A_a^bf=\alpha(\{ (x,y)\colon a\leq x\leq b \mbox{ and } 0\leq y\leq f(x)\}).$$

8.3   Theorem (Integrals of power functions.) Let $r\in\mbox{{\bf Q}}$, and let $a,b$ be real numbers such that $0<a\leq b$. Let $f_r(x) = x^r$ for $a \leq x \leq b$. Then

$$\int_a^b f_r = \cases{ \displaystyle {{b^{r+1} - a^{r+1} \ove... ...setminus \{-1\}$\vspace{1ex}\cr \ln(b) - \ln(a) & if $r=-1$.}$$

In general integrable functions may take negative as well as positive values and in these cases $${\int_a^bf}$$ does not represent an area.

The next theorem shows that monotonic functions are integrable even if they take on negative values.

8.4   Example (Monotonic functions are integrable II.) Let $f$ be a
monotonic function from an interval $[a,b]$ to $\mbox{{\bf R}}$. Let $B$ be a non-positive number such that $f(x)\geq B$ for all $x\in [a,b]$. Let $g(x)=f(x)-B$.

Then $g$ is a monotonic function from $[a,b]$ to $\mbox{{\bf R}}_{\geq 0}$. Hence by theorem 7.6, $g$ is integrable on $[a,b]$ and $${\int_a^bg=A_a^b(g)}$$. Now let $\{P_n\}$ be a sequence of partitions of $[a,b]$ such that $\{\mu (P_n)\}\to 0$, and let $\{S_n\}$ be a sequence such that for each $n$ in $\mbox{${\mbox{{\bf Z}}}^{+}$}$, $S_n$ is a sample for $P_n$. Then

$$\{\sum (g,P_n,S_n)\}\to A_a^b(g).$$ (8.5)

If $P_n=\{x_0,\cdots ,x_m\}$ and $S_n=\{s_1,\cdots ,s_m\}$ then

$$\begin{eqnarray*} \sum (g,P_n,S_n) &=& \sum_{i=1}^m g(s_i)(x_i-x_{i-1})\\ &=& \... ...-1}) - B\sum_{i=1}^m(x_i-x_{i-1})\\ &=&\sum (f,P_n,S_n)-B(b-a). \end{eqnarray*}$$

Thus by (8.5)

$$\{\sum (f,P_n,S_n)-B(b-a)\}\to A_a^b(g).$$

If we use the fact that $\{B(b-a)\}\to B(b-a)$, and then use the sum theorem for limits of sequences, we get

$$\{\sum (f,P_n,S_n)\}\to A_a^b (g)+B(b-a).$$

It follows from the definition of integrable functions that $f$ is integrable on $[a,b]$ and

$$\int_a^bf=A_a^b(g)+B(b-a)=\int_a^bg+B(b-a)=\int_a^bg-\vert B\vert(b-a).$$

Thus in figure b, $${\int_a^bf}$$ represents the shaded area with the area of the thick box subtracted from it, which is the same as the area of the region marked ``$+$" in figures c and d, with the area of the region marked ``$-$" subtracted from it.

The figure represents a geometric interpretation for a Riemann sum. In the figure

$$f(s_i)>0 \mbox{ for } i=1,2,3,\quad f(s_i)<0 \mbox{ for } i=4,5.$$

$$\sum_{i=1}^3 f(s_i)(x_i-x_{i-1})$$

is the area of $${\bigcup_{i=1}^3 B\Big(x_{i-1},x_i\colon 0,f(s_i)\Big)}$$ and

$$\sum_{i=4}^5f(s_i)(x_i-x_{i-1})$$

is the negative of the area of

$$\bigcup_{i=4}^5 B(x_{i-1},x_i:f(s_i),0).$$

In general you should think of $${\int_a^bf}$$ as the difference $\alpha (S^+)-\alpha (S^-)$ where

$$S^+=\{(x,y)\colon a\leq x\leq b \mbox{ and } 0\leq y\leq f(x)\}$$

and

$$S^-=\{(x,y)\colon a\leq x\leq b \mbox{ and } f(x)\leq y\leq 0\}.$$

8.6   Exercise. The graphs of two functions $f,g$ from $[0,2]$ to $\mbox{{\bf R}}$ are sketched below.

Let

$$F(x)=\Big( f(x)\Big)^2 \mbox{ for } 0\leq x\leq 2,\quad G(x)=\Big( g(x)\Big)^2 \mbox{ for } 0\leq x\leq 2.$$

Which is larger:

a)

$${\int_0^1f}$$ or $${\int_0^1 F}$$?

b)

$${\int_0^1 g}$$ or $${\int_0^1 G}$$?

c)

$${\int_0^1f}$$ or $${\int_0^1 g}$$?

d)

$${\int_0^{1/2}g}$$ or $${\int_0^{1/2}G}$$?

e)

$${\int_0^2 g}$$ or $${\int_0^2 G}$$?

Explain how you decided on your answers. Your explanations may be informal, but they should be convincing.

8.7   Exercise. Below is the graph of a function $g$. By looking at the graph of $g$ estimate the following integrals. (No explanation is necessary.)

Graph of $g$

a)

$${ \int_{1\over 4}^{3\over 4} g}$$.

b)

$${ \int_1^2 g}$$.

c)

$${ \int_0^{3\over 4} g}$$.

8.8   Exercise. Sketch the graph of one function $f$ satisfying all four of the following conditions.

a)

$${ \int_0^1 f = 1}$$.

b)

$${ \int_0^2 f = -1}$$.

c)

$${ \int_0^3 f = 0}$$.

d)

$${ \int_0^4 f = 1}$$.